The most prominent advantage of employing PAs is that they allow for a
systematic and model-independent analysis. Moreover, which is
especially relevant to our work, they can reproduce analytic features of the
function they are approximating, such as poles, residues and zeros. In
addition, in some cases, the use of these approximants is validated by
convergence theorems. The most important one for this work is Pommerenke's
theorem, which states that PAs of the sequence \(P_N^{N+k}(z)\) , for fixed \(k\) ,
applied to a meromorphic function \(f(z)\) , converges to \(f(z)\)  for
\(N\rightarrow \infty \)  [1]}, [2]}. On the other hand,
this theorem predicts that the Padé can have extraneous poles, which
cannot be identified as singularities of the original function and
move away when the order of the PA is increased. There are also spurious
poles that can appear close to zeros, which are the defects or
Froissart doublets [1]}, [2]}, [5]} — these defects are of transient nature and should
also disappear for PAs of higher order. It is important to
stress that approximants with such defects can still be employed
to study the considered function, away from these singularities.
